Strongly singular convolution operators on modulation spaces (Q2891001)

The modulation space \(M_s^{p,q}\) is defined by the norm NEWLINE\[NEWLINE\|f\|_{M_s^{p,q}} =\Bigl(\int_{\mathbb{R}^n}\Bigl(\int_{\mathbb{R}^n} |V_gf(x,w)|^p \, dx\Bigr)^{q/p} (1+|w|^2)^{sq/2}\, dw\Bigr)^{1/q} NEWLINE\]NEWLINE with suitable modifications for \(p=\infty\) or \(q=\infty\). Here \(V_g f\) is the short-time Fourier transform \( V_g (f)(x,\xi)=\langle f,\, M_\xi T_x g\rangle =\int e^{-2\pi i \xi\cdot y} f(y)\overline{g}(y-x)\, dy\) for a suitable window function \(g\) such as the Gaussian. A strongly singular convolution operator has the form \(T(f)(x)=(K\ast f)(x)\), where \(f\) is a suitable test function and \(K\) is the kernel \(K(x)=e^{i|x|^a}/|x|^\alpha\), \(x\neq 0\), where \(a\neq 1\) and \(0<\alpha<n\). The main result of this work is that if \(a>0\), \(a\neq 1\), and \(\alpha<n\) then the operator \(T\) is bounded on \(M^{p,q}_s(\mathbb{R}^n)\) for \(0<p\leq \infty\), \(0<q\leq \infty\), and \(s\in \mathbb{R}\). A boundedness result on the weighted modulation space for the strongly singular operator with kernel \(K_\alpha =e^{i|x|^{-a}}/|x|^{n+\alpha}\) is also proved. In this case, \(T\) is bounded on \(M^{p,q}_s(\mathbb{R}^n)\) with \(p\in [1,\infty]\), \(q\in (0,\infty]\) and \(s\in\mathbb{R}\) provided that \(0<\alpha\leq na/2\). Finally, multiplier results are also proved. Set NEWLINE\[NEWLINEH_{bc}=\sum_{n=-\infty}^\infty c_n (M_{bn} H M_{-bn}-M_{b(n+1)} H M_{-b(n+1)}),NEWLINE\]NEWLINE where \(H\) denotes the Hilbert transform and \(M\) is the modulation operator \(M_b (f)(t)=e^{2\pi i b\cdot t} f(t)\). Then \(H_{bc}\) is bounded on the same range of modulation spaces if \(c=\{c_n\}\in\ell^1\). One can extend to \(0<p<1\) if \(c\in\ell^p\). The work builds on earlier work of a number of authors concerning the action of such operators on \(L^p\)-spaces (e.g., [\textit{P. Sjölin}, Indiana Univ. Math. J. 30, 47--55, (1981; Zbl 0419.47020)]), and of \textit{K. A. Okoudjou} et al. [J. Funct. Anal. 246, No. 2, 366--384 (2007; Zbl 1120.42010)] regarding Fourier multipliers on modulation spaces.